Properties

Label 4-21e4-1.1-c1e2-0-12
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $12.4002$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 14·13-s − 7·19-s + 5·25-s − 7·31-s + 37-s + 10·43-s + 28·52-s + 14·61-s − 8·64-s − 11·67-s − 7·73-s − 14·76-s + 13·79-s − 28·97-s + 10·100-s − 7·103-s − 17·109-s + 11·121-s − 14·124-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 4-s + 3.88·13-s − 1.60·19-s + 25-s − 1.25·31-s + 0.164·37-s + 1.52·43-s + 3.88·52-s + 1.79·61-s − 64-s − 1.34·67-s − 0.819·73-s − 1.60·76-s + 1.46·79-s − 2.84·97-s + 100-s − 0.689·103-s − 1.62·109-s + 121-s − 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.164·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(12.4002\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.558692321\)
\(L(\frac12)\) \(\approx\) \(2.558692321\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.12860371615261467851924439168, −10.90353487336046868288197811939, −10.70602380960564718246505483983, −10.36540667070145818476042774431, −9.283608278934972193511150123357, −9.103973899798186914167211511003, −8.467170440399022191272434916402, −8.397336184619730793573480375057, −7.72728847174516648442637002954, −7.00540619073477057371705526330, −6.55084563165038723035480240860, −6.33877706044315874331776533094, −5.79882880499407263774288320665, −5.39149127483252496338818142677, −4.15021255882218326662270793415, −4.09192514111419180474122829763, −3.36028575022800212308720570501, −2.64949038910737600429711607958, −1.78678407936487485488284490821, −1.15061253936147325347589354896, 1.15061253936147325347589354896, 1.78678407936487485488284490821, 2.64949038910737600429711607958, 3.36028575022800212308720570501, 4.09192514111419180474122829763, 4.15021255882218326662270793415, 5.39149127483252496338818142677, 5.79882880499407263774288320665, 6.33877706044315874331776533094, 6.55084563165038723035480240860, 7.00540619073477057371705526330, 7.72728847174516648442637002954, 8.397336184619730793573480375057, 8.467170440399022191272434916402, 9.103973899798186914167211511003, 9.283608278934972193511150123357, 10.36540667070145818476042774431, 10.70602380960564718246505483983, 10.90353487336046868288197811939, 11.12860371615261467851924439168

Graph of the $Z$-function along the critical line